Embedded newform 243.2.i.a.103.24

• Label: 243.2.i.a.103.24
• Level: $243$
• Weight: $2$
• Character: 243.103
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.i (of order \(81\), degree \(54\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(1404\)
Relative dimension:	\(26\) over \(\Q(\zeta_{81})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{81}]$

Embedding invariants

Embedding label			103.24
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.26585 + 0.630742i) q^{2} +(-0.366995 + 1.69272i) q^{3} +(3.02391 + 1.82494i) q^{4} +(-2.74419 + 0.106487i) q^{5} +(-1.89923 + 3.60398i) q^{6} +(2.01284 + 0.314803i) q^{7} +(2.47257 + 2.62077i) q^{8} +(-2.73063 - 1.24244i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{81}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.26585	+	0.630742i	1.60220	+	0.446002i	0.950052	−	0.312091i	\(-0.101030\pi\)
							0.652146	+	0.758094i	\(0.273869\pi\)
\(3\)	−0.366995	+	1.69272i	−0.211884	+	0.977295i
							\(5\)	−2.74419	+	0.106487i	−1.22724	+	0.0476224i	−0.644222	−	0.764839i	\(-0.722819\pi\)
−0.583015	+	0.812461i	\(0.698127\pi\)
\(7\)	2.01284	+	0.314803i	0.760783	+	0.118984i	0.522998	−	0.852334i	\(-0.324813\pi\)
							0.237785	+	0.971318i	\(0.423579\pi\)
\(11\)	0.475940	+	3.48450i	0.143501	+	1.05062i	0.911832	+	0.410563i	\(0.134668\pi\)
							−0.768331	+	0.640053i	\(0.778912\pi\)
\(13\)	3.06939	−	4.47529i	0.851297	−	1.24122i	−0.117427	−	0.993081i	\(-0.537465\pi\)
							0.968724	−	0.248140i	\(-0.0798192\pi\)
\(17\)	6.06937	−	3.04815i	1.47204	−	0.739286i	0.481491	−	0.876451i	\(-0.340095\pi\)
							0.990548	+	0.137165i	\(0.0437991\pi\)
\(19\)	−0.238916	−	4.10203i	−0.0548111	−	0.941070i	−0.907869	−	0.419254i	\(-0.862292\pi\)
							0.853058	−	0.521816i	\(-0.174745\pi\)
\(23\)	−3.06012	+	7.92590i	−0.638079	+	1.65267i	0.113912	+	0.993491i	\(0.463662\pi\)
							−0.751990	+	0.659174i	\(0.770906\pi\)
\(29\)	−3.45486	−	2.46939i	−0.641552	−	0.458554i	0.213607	−	0.976920i	\(-0.431479\pi\)
							−0.855159	+	0.518366i	\(0.826541\pi\)
\(31\)	−3.03787	+	3.48101i	−0.545617	+	0.625208i	−0.958334	−	0.285652i	\(-0.907790\pi\)
							0.412716	+	0.910860i	\(0.364580\pi\)
\(37\)	−1.71554	−	2.30438i	−0.282034	−	0.378837i	0.638420	−	0.769688i	\(-0.279588\pi\)
							−0.920454	+	0.390851i	\(0.872181\pi\)
\(41\)	0.270732	−	1.05105i	0.0422813	−	0.164146i	−0.943765	−	0.330617i	\(-0.892743\pi\)
							0.986046	+	0.166471i	\(0.0532371\pi\)
\(43\)	2.62829	+	2.38505i	0.400811	+	0.363717i	0.847338	−	0.531053i	\(-0.178204\pi\)
							−0.446527	+	0.894770i	\(0.647339\pi\)
\(47\)	−1.07362	−	1.23024i	−0.156604	−	0.179448i	0.669772	−	0.742567i	\(-0.266392\pi\)
							−0.826376	+	0.563118i	\(0.809602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.24	✓	1404
3.2	odd	2		729.2.i.a.685.3		1404
243.92	odd	162		729.2.i.a.613.3		1404
243.151	even	81	inner	243.2.i.a.151.24	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.24	✓	1404	1.1	even	1	trivial
243.2.i.a.151.24	yes	1404	243.151	even	81	inner
729.2.i.a.613.3		1404	243.92	odd	162
729.2.i.a.685.3		1404	3.2	odd	2